1054. Hanoi Tower

Time limit: 1.0 secondMemory limit: 64 MB

Background

"Hanoi tower" puzzle is well known. There are 3 pegs with numbers: #1, #2 and #3. N disks of different diameters are set on the first peg in the following order: the lower disk is set, the larger diameter it has. Your aim is to move all disks onto the second peg using the peg #3 as an auxiliary one. Following the rules within a move it's allowed to replace only one uppermost disk. Besides it's forbidden to put the disk of the bigger diameter onto the disk of the smaller one.

Distribution of disks on the pegs during the game must be assigned as sequence D (element #i of the sequence is equal to the number of peg where the disk #i is placed on). For instance, game status after the third move is apparently determined by sequence D=(3, 3, 1) (the first and the second disks are set on the third peg and the third disk is on the peg #1).

Example. Let's assume that 3 disks are set on the peg #1. Then the movement of the disks can be depicted in the following table (disks are numbered in ascending order of diameters):

Point of move

Peg #1 disks

Peg #2 disks

Peg #3 disks

sequence D

0

1, 2, 3

-

-

1, 1, 1

1

2, 3

1

-

2, 1, 1

2

3

1

2

2, 3, 1

3

3

-

1, 2

3, 3, 1

4

-

3

1, 2

3, 3, 2

5

1

3

2

1, 3, 2

6

1

2, 3

-

1, 2, 2

7

-

1, 2, 3

-

2, 2, 2

Problem

Your aim is either to determine (using arbitrary sequence D) the number of moves from the beginning of the game to the given position on condition of performing the optimal algorithm; or to print "−1" in the case of incorrect sequence declaration (i.e. the given position cannot be achieved performing the optimal algorithm).

Reference notes. Optimal algorithm can be specified with the following recursive procedure.